Remark 5.8 By Proposition REF  the central element \(\Delta _{2n+2}^2\)  belongs to the kernel of \(\bar{f}\) .
It can be related to our special element \(\alpha _n\)  as follows. By [1]} we have
\(\Delta _{2n+2}^2 = \Delta _{2n+1}^2 \, A_{1,2n+2} A_{2,2n+2} \cdots A_{2n+1,2n+2} \, .\) 
